How do I use Windows 10 built-in package manager?

You were close. First, you have to set the execution policy to allow scripts, otherwise it'll silently fail while reporting success (bug):

Set-ExecutionPolicy RemoteSigned

Both the package provider (Chocolatey plugin) and package source (URL to specific Chocolatey repo) need to be installed/registered with PackageManagement. Get-PackageProvider with the -Force flag causes it to bootstrap, which apparently takes care of both (more in the help about -Force):

Get-PackageProvider Chocolatey -Force | Out-Null

Then I can search for the package:

Find-Package vlc -Force

Name       Version          Source           Summary
----       -------          ------           -------
vlc        2.2.1.20150630   chocolatey       VLC Media Player

And install it (-Force so it doesn't prompt for confirmation):

Install-Package vlc -Force | Out-Null